A dynamic model of bidding patterns in sponsored search auctions

Inf Technol Manag (2011) 12:1–16 DOI 10.1007/s10799-010-0077-y

A dynamic model of bidding patterns in sponsored search auctions Kursad Asdemir

Published online: 9 December 2010 Springer Science+Business Media, LLC 2010

Abstract We develop an infinite horizon alternativemove model of the unique second-price sponsored search auction. We use this model to explain two distinguishable bidding patterns observed in our bidding data: bidding war cycle and stable bid. With examples, we show that only a small portion of the value generated may be extracted by search engines, if advertisers are engaged in bidding war cycles. Finally, we show the impact of auction design on advertiser bids and search engine revenue. Keywords Sponsored search auction Dynamic oligopoly Tacit collusion E-commerce

1 Introduction In sponsored search auctions, advertisers relentlessly bid for clicks 24 hours a day, 7 days a week all year long. At any moment, hundreds of thousands of auctions are under way. Fierce competition can ensue for valuable keywords. Some advertisers are willing to pay as high as $100 for a single click for certain keywords that may create lucrative

Previous versions of this paper have been presented at The University of Texas at Dallas, University of Alberta, Yahoo! Research Labs, University of British Columbia, CORS/INFORMS Joint International Meeting 2004, Alberta IO Conference 2005. I thank all the participants. I acknowledge special contributions of Varghese S. Jacob, Nanda Kumar, and David Pennock to this paper. K. Asdemir (&) College of Business, American University of the Middle East, P.O. Box 220, 15453 Dasman, Kuwait e-mail: [email protected]

business opportunities [2]. This fascinating marketplace is the foundation of a vibrant sponsored search industry and one of the engines that powers the Internet economy. To exploit this competition for good positions, search engines use auctions as a pricing and ranking mechanism. Advertisers bid on each potential keyword that could be relevant for their product. A separate sponsored search auction is held for each keyword, but unlike a normal auction, say for a piece of art, which takes place once and for all, the auctions for the keywords take place continuously and never close. In other words, when an advertiser submits a bid, she is allocated a position based on the ranking scheme of the search engine and this position may change depending on the opponents’ bids and the market conditions. Thus, advertisers bid for revenue flows as opposed to a single item or multiple items. There have been two major ranking and payment schemes. The first ranking scheme is to rank advertisers based only on their bids. The bids are available to all auction participants and to the public. This is the Overture type auction (later acquired by Yahoo! and renamed as Yahoo! Search Engine Marketing). In the initial design of this auction, advertisers used to pay their bids (first-price auction). In 2002, Overture switched to a second-price variant following Google where advertisers pay the next advertiser’s bid plus one cent [10]. The second scheme is introduced by Google in which ranking is based on an advertiser’s quality score which was initially click rate times the bid, that is, the total revenue collected by Google from that advertiser. Google, then, extended this score to include several other variables such as the quality of the landing page a visitor arrives after clicking on a sponsored link. Google does not disclose bids even to the advertisers. As a result of the success of the Google-type auction, Yahoo! announced the overhaul of its system [8]. In this

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$2.20

Inf Technol Manag (2011) 12:1–16 Bid 1st position bids

$2.10

2nd position bids

Bid

$17.50

1st position bids

$17.45

2nd position bids

$17.40

$2.00

$17.35

$1.90

$17.30 $1.80

$17.25

$1.70

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$1.60

$17.15

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$1.40 1

241 481 721 961 1201 1441 1681 1921 2161 2401 2641

$17.00

Observation Number Fig. 1 First and second position bids for ‘‘notebook computer’’ in 15 min intervals from 12/30/2003 16:38–1/26/2004 20:09 Source: http://uv.bidtool.overture.com/d/search/tools/bidtool/

paper, we are concerned with the Overture type auction with public bids and second-price payment scheme. Since a sponsored search auction never closes and there is no apparent winner, advertisers have developed dynamic bidding strategies that take into account the impact of their bids on the strategic bidding behavior of opponents. Although there is continuous experimentation and learning by the advertisers, bidding data from Yahoo! reveals two clear bidding patterns. One pattern, which is shown in Fig. 1, is a bidding war cycle (BWC hereafter).1 Advertisers outbid each other until one of them drops their bid and the other one follows by dropping her bid to just 1¢ above the competitor’s bid. The other pattern, which can be seen in Fig. 2, is that of a stable set of bids for a relatively long period of time. Table 1 shows different bidding patterns for computerrelated keywords in January 2004 in Overture auctions. Although advertisers use multiple bidding strategies in all of the auctions, we can clearly recognize a bidding pattern in the majority of the auctions. In Table 1, the main determinant of observing BWCs is the existence of the major computer manufacturers (dell4me.com, gateway.com, shoptoshiba.com, sonystyle.com). If the keyword is too specialized like Dell computer or Apple computer, we observe stable bids. In addition, if there are other advertisers competing for clicks such as the generic phrase computer, the major players cannot form BWCs. This finding shows the importance of BWCs. BWCs may not be specific to individual keyword auctions but major companies in an industry may adopt such strategies. Because advertisers’ bids reflect the nature of competition, the existence of recognizable bidding patterns begs the question: ‘‘What do 1

We want to emphasize that the series in all the graphs in this paper are discrete series. However, we connect the individual data points together (draw a line graph) to help the reader understand how bids change over time.

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280

559

838 1117 1396 1675 1954 2233 2512

Observation Number Fig. 2 First and second position bids for ‘‘erp software’’ in 15 min intervals from 12/30/2003 16:59–1/26/2004 17:39 Source: http://uv.bid tool.overture.com/d/search/tools/bidtool/

Table 1 Bidding patterns in top 2 to 4 positions in computer related phrases in January 2004 in Overture auctions Pattern

Keywords

Bidding war cycle Notebook computer, notebook, laptop computer, (BWC) laptop, desktop, buy computer, computer sales, computer shopping Stable bid

Dell computer, apple computer, PC, used computer

Not classified

Computer, desktop computer, cheap computer

these different bidding patterns suggest about advertiser behavior?’’ In contrast to the majority of the literature on sponsored search auctions, we adopt a dynamic oligopoly approach for the Overture-type auction. We develop an infinite sequential-move duopoly game where advertisers commit to their bids for two periods. One advertiser sets her bid at odd periods and the other sets her bid at even periods. This setup captures the institutional feature that advertisers observe each other’s bid before they decide on their bid.2 Bids are selected from a discrete grid. We show that BWC and stable bid patterns can result from a symmetric Markov perfect equilibrium.3 We obtain several results and insights on dynamic bidding behavior in sponsored search auctions. We propose several options for search engines to increase their revenues. These include introducing a reserve price for the first two positions, making the second position less 2

The structure of this game is due to Cyert and DeGroot [3]. See Fudenberg and Tirole [7] Chapter 13 for a discussion of Markov-perfect equilibrium. Our paper is related to Maskin and Tirole [12]’s model of price competition in an infinite duopoly. They present an alternative-move model, which can generate price war cycle on its equilibrium path. Price war cycles have been reported especially in retail gasoline markets [4].

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desirable for advertisers, and increasing the grid size (minimum bid increment). Mechanism design/auction theory treatments of sponsored search auctions derive the equilibrium of a oneperiod bidding game (e.g. [1, 6, 14]). Because these papers model sponsored search auctions as one period games, advertisers set their bids only once. This literature stream suggests a stable bid pattern that may explain the bidding behavior in Fig. 2. Our study develops a dynamic model where advertisers set their bids alternatively. This allows advertisers to set their bids by considering their expectations of how bids will change in the future. Studies of BWCs do not present a formal theoretical treatment of the characteristics of these cycles. A dynamic model of BWCs is important for at least three reasons. First, Edelman and Ostrovsky [5] suggest that BWCs occur only in first-price auctions because of proxy bidding. We show that BWCs occur even in second-price sponsored search auctions. Second, if a search engine wants to assess the level of competition from bidding data, it needs to infer the valuations of bidders. For example, Edelman and Ostrovsky [5] present such a study for the first-price Overture auction. However, Edelman and Ostrovsky [5] conjecture theoretically that advertisers would drop their bids when the bids reach the minimum between two advertisers. We show that this conjecture may not hold even in a second-price auction. Third, advertisers may be interested in devising strategies that would keep the bids low. Kitts, Laxminarayan, and LeBlanc [9] suggest that advertisers should engage in BWCs and outbid each other up to the Nash-equilibrium bids, and then they would drop their bids. Again, we show that advertisers can sustain much lower dynamic equilibrium bids. The rest of this paper is organized as follows: Sect. 2 presents the basic setup of the model. Section 3 characterizes an equilibrium that can generate stable bids, while in Sect. 4 we turn our attention to the BWC equilibrium. Sect. 5 studies the impact of changing the auction design parameters on a search engine’s revenue. Finally, we conclude the paper in Sect. 6.

2 The model 2.1 The sponsored search auction We consider an Overture-type auction for a single keyword such as ‘‘notebook computer’’ (Fig. 2). There are two available slots in the sponsored listings area of the results page. There are two strategic symmetric advertisers, advertiser X and advertiser Y. We model the bidding game between X and Y. The sponsored listings are sorted in descending order according to the submitted bids when a

3

search engine user submits a query. Payment is made to the search engine when a user clicks on a sponsored listing. Bids are selected from a discrete grid with a grid size (minimum bid increment) k. A smaller k represents a finer grid. For example, Google and Yahoo! use a grid size of 1¢, while Business.com used to have a grid size of 10¢ in their respective sponsored search auctions.4 We assume that there is a minimum bid l and the advertisers prefer to participate in the auction by bidding l and occupying the second position. Accordingly, the bids of X and Y can only take values l, l ? k, l ? 2 k,… and so forth. In order to define the ranking rule and payment function, we need to represent the advertiser’s bid vector at different time periods. We define a reduced state bXt ; bYt at time period t, where bXt and bYt are the bids of the advertisers X and Y, respectively.5 In the dynamic game, the full state also includes the advertiser who is about to move. In the steady state of the dynamic game and the one period game, the time period t does not affect the payoffs of the advertisers and the search engine, therefore we omit the time index t in several expressions. In the Overture-type auction, advertisers are ranked according to only advertisers’ bids. Since the Overture auction is a second-price auction, the payment for a click at the first position depends on the advertiser’s bid at the second position. The first-position advertiser pays k more than the second advertiser’s bid, if her bid is greater than the second advertiser’s bid. The second-position advertiser always pays l. Let rX (rY) be the position of X (Y). We define the payment function at state (bX, bY) as: 8 Y b þ k if bX [ bY ; > > > > < bY if bX ¼ bY and r X ¼ 1; ð1Þ C X bX ; bY ¼ > l if bX ¼ bY and r X ¼ 2; > > > : l if bX \bY : 2.2 Advertisers We now define the payoff function for the advertisers. Each advertiser at the jth position gets aj number of clicks at every period and gets a value of sj per click, for j = 1, 2. We can interpret the value sj as average profits generated via conversions of users clicking on sponsored listings into buyers. Let P1(b) be the payoff in the first position with payment b and P2 be the payoff in the second position. In the first position, X makes a1s1 and pays a1b. In the second position X makes a2s2 and pays a2l. Then, P1 ðbÞ ¼ a1 ðs1 bÞ and P2 ¼ a2 ðs2 lÞ: 4

http://www.business.com/info/common_questions/what_business_ costs.asp (Accessed: 6/5/2006). 5 We model a second-price auction that makes a two-dimensional state space necessary for constructing equilibrium reaction functions.

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Given the payment function in (1), we can write X’s per period payoff function at state (bX, bY) as: 8 P1 bY þ k if bX [ bY ; > > > > < P 1 bY if bX ¼ bY and r X ¼ 1; X X Y ð2Þ P b ;b ¼ > P2 if bX ¼ bY and r X ¼ 2; > > > : P2 if bX \bY : We can define Y’s payoff function similarly. In the dynamic game, we assume that advertisers discount future cash flows by the discount factor d and maximize their value of discounted profits. Hence, X’s objective function is: 1 X ^X ¼ max P dt PX bXt ; bYt : t¼0

2.3 The equilibrium concept We proceed with the description of the dynamic game and the equilibrium concept—Markov perfect equilibrium (MPE). Then, we identify the conditions that must be satisfied for a strategy profile to constitute an MPE. We model the dynamic game as an infinite-horizon discrete-time game. The unique feature of this game is that each advertiser determines her bid in alternate periods sequentially after observing the opponent’s bid. That is, advertiser X sets her bid at the beginning of periods 0, 2, 4, …, 2t–2, 2t; while advertiser Y sets her bid at the beginning of periods 1, 3, 5, …, 2t–1, 2t ?1, and so on. Advertisers cannot change their bids for two periods.6 This setup captures the idea that the advertisers commit to their new bids for two periods while waiting one period before reacting to the opponent’s bid. For example, if Y’s initial bid is $1.49 and X’s new bid is $1.50 at time 0 ðbX0 ¼ $1:50Þ then X is committed to set her bid $1.50 in the next period ðbX1 ¼ $1:50Þ and likewise if bY1 ¼ $1:51 then bY2 ¼ $1:51: In this example, after observing X’s bid of $1.50, Y sets her bid to $1.51 and obtains the first position. Consequently, when an advertiser sets her bid, she knows the current bids, the current ranking of listings, and thus the exact position she is going to occupy and the new payment for every possible value of her new bid. In this setting, we characterize the advertisers’ strategies with static Markov reaction functions Ri(bX, bY), which

6

The two-period commitment assumption may seem arbitrary, since advertisers can change their bids at anytime. Maskin and Tirole [11] show that two-period commitment model is equivalent to a model in which the advertisers wait a random amount of time, which is exponentially distributed, before reacting to each other’s bids.

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represents advertiser i’s new bid at state (bX, bY) for i = X, Y. Note that Ri(bX, bY) is static in that it only depends on the current state but not the current time. In the above example, RY($1.50, $1.49) = $1.51. Reaction functions are not limited to deterministic strategies; advertisers can also use mixed strategies. In other words, Ri(bX, bY) can represent a probability distribution over possible bid values. A strategy profile is the vector of static reaction functions of both advertisers X and Y: {RX, RY}. A strategy profile is an equilibrium strategy profile if RX is an optimal response to RY and vice versa. We now discuss these conditions for an MPE. MPE strategies only depend on the payoff-relevant history [11]. As defined above, the reaction functions are Markov, i.e. they only depend on the current state. Additionally, MPE is perfect, i.e. in an MPE every subgame of the dynamic game is also an MPE. These properties of MPE calls for a dynamic programming approach. To specify MPE conditions, we define a pair of value functions for each advertiser at each state. We define the value function VX,X(bX, bY) as the expected discounted total future profit of advertiser X when advertiser X is about to move at state (bX, bY) and VX,Y(bX, bY) as X’s value function when advertiser Y is about to move at state (bX, bY). Corresponding definitions apply for advertiser Y. It follows that the necessary and sufficient conditions for these value functions and reaction functions to form an MPE are: V X;X bX ; bY ¼ max PX b^X ; bY þ dV X;Y b^X ; bY ð3Þ b^X

V X;Y bX ;bY ¼ ERY ðbX ;bY Þ PX bX ; b^Y þdV X;X bX ; b^Y

ð4Þ

where RX(bX, bY) is the choice of b^X that maximizes the objective function in (3) and the expectation in (4) is taken over b^Y which belongs to the support of RY(bX, bY). We are now ready to study strategy profiles that satisfy the MPE conditions (3) and (4).

3 Stable-bid equilibrium We provide a possible explanation of the observed stable bid pattern of Fig. 2. It is obvious that advertisers do not change their bids for long periods in Fig. 2. This behavior may be driven by advertiser’s belief that it is not worth the effort to continuously track and update their bids. Then, the interesting question is ‘‘How should an advertiser set her bid, if she does not want to make frequent bid changes?’’ In Proposition 1, we show that if advertisers play their weakly dominant strategies, it will be an MPE. This implies that advertisers may not alter their bids unless the underlying cost and revenue parameters change significantly.


5

Fig. 3 Normal form of the simultaneous move game in Lemma 1: The row (column) player is X (Y)

l l

Π1(l)

l+k

Π1(l+k)

…

Π1(l+k)

b*-k

Π1(l+k)

Π2 Π2

b*

Π1(l+k)

b*+k

Π1(l+k)

…

Π1(l+k)

l+k

Π2 Π2 Π2 Π2 Π2 Π2

P1 ðb Þ P2 [ P1 ðb þ kÞ:

Π1(l+k) Π2 Π1(l+2k) …

…

Π1(b*-k)

…

Π1(b*)

Π1(l+2k) Π2 Π1(l+2k) Π2

…

Π1(b*)

*

Lemma 1 The strategy profile (b*, b*) is the unique iterated-dominance equilibrium of the one-period simultaneous move game. Proof of Lemma 1 Since advertisers are symmetric, we drop the advertiser index in the payoff functions; and we assume X takes the first position when bids are equal. We solve for iterated-dominance equilibrium by the iterated elimination of weakly dominated strategies [13]. Consider the normal form of this game in Fig. 3, where X is the row player and Y is the column player. X’s payoff is given on the first line while Y’s payoff is given on the second line in each cell for the corresponding bids. For the column player Y, it is easily seen that l ? k weakly dominates l since P1(l ? k) [ P2; l ? 2 k weakly dominates l ? k since P1(l ? 2 k) [ P2….b* weakly dominates b*- k since P1(b*) C P2. b* weakly dominates b* ? k since P1(b* ? k) \ P2. b* ? k weakly dominates b* ? 2 k since P1(b* ? 2 k) \ P2… Therefore, b* weakly dominates every other strategy for Y. Elimination of the Rasmusen [13], page 23.

Π2

…

Π1(b*)

Π2 Π2 Π2

b* Π2

b*+k Π2

… Π2

Π1(l+k) Π2 Π1(l+2k) …

Π1(l+k) Π2 Π1(l+2k) …

Π1(l+k) Π2 Π1(l+2k) …

Π2

Π2

Π2

Π1(b*) Π1(b*) Π2 Π1(b*+k) Π2 Π1(b*+k) Π2

Π2 Π1(b*+k) Π1(b*+k) Π2 Π1(b*+2k) Π2

Π1(b*)

Π1(b*) Π2 Π1(b*+k) Π2 Π1(b*+2k) …

weakly dominated strategies leaves only b* for Y. In the remaining game, X plays b*, since P1(b*) C P2. Q.E.D. In order to show that the symmetric strategy in Proposition 1 forms an MPE, we show that there is no one-period profitable deviation [11]. 1.

2.

ð5Þ

Proof of Proposition 1 The proof has two parts. We, first, characterize the iterated-dominance equilibrium7 of the one-period game in Lemma 1. After characterizing (b*, b*) in the one-period game, we will show that it also constitutes an equilibrium in the dynamic game.

7

Π2

Π1(l+k) Π2 Π1(l+2k) …

where the strategy profile (b , b ) is the unique iterateddominance equilibrium of the one-period game in which advertisers set their bids simultaneously. b* is the highest payment on the bid grid that makes an advertiser weakly better off in the first position than in the second position and satisfies

Π2

Π1(l+k) Π1(l+k) Π2 Π1(l+2k) Π2 Π1(l+2k) Π2 Π1(l+2k) Π2

Proposition 1 The following strategy profile {R, R}, R = RX(bX, bY) constitutes a Markov Perfect Equilibrium with RX b X ; bY ¼ b ; *

b *− k

…

3.

For bidding against bY \ b*, the equilibrium reaction RX(•, bY) = b* is not dominated, since P1(bY ? k) C P1(b*) C P2 from condition (5). For bidding against bY [ b*, RX(•, bY) = b* is not dominated, since from (5) P1(bY ? k) \ P1(b* ? k) \ P 2. For bidding against bY = b*, RX(•, bY) = b* is not dominated, since P1(b* ? k) \ P2. Q.E.D.

The equilibrium bid b* can be characterized as follows. If an advertiser, say X, bids too high (bX [ b*), she earns less than the second position payoff when the opponent Y sets her bid at bX-k. On the other hand, if X bids too low (bX \ b*), she may lose the opportunity of being at the first position when the opponent Y sets a lower bid than b* (bX \ bY \ b*). Intuitively, b* minimizes an advertiser’s potential loss if she is competing against a vindictive bidder (an opponent setting her bid one increment (k) below the advertiser’s bid, see [15]); at the same time b* maximizes the possibility of attaining the first position. In other words, b* is the bid that makes an advertiser’s worst-case first position payoff weakly better than her second position payoff. In the proof of Proposition 1 above, we show b* weakly dominates all other bids. Therefore, in the one period game, both advertisers end up bidding b*. This very much hurts the first position advertiser because she ends up paying the highest amount she is willing to pay. If she pays more than b* for the first position, she would prefer the second position. Noting that the minimum bid is always l, we can characterize the equilibrium of Proposition 1 as the equilibrium that is very favorable to the search engine. What drives this result is the following. In a sponsored search auction with stable bids, advertisers set their bids for long periods

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(a month or a few weeks in our dataset) without monitoring other advertisers’ bids. During this period, other advertisers may join the auction or existing advertisers may change their bids. This means that the advertisers decide their bids in the absence of other advertisers’ bids. Therefore, the one-period game of Lemma 1, in which an advertiser sets her bid without the knowledge of the opponent’s bid, may be a good approximation of this setting. This prevents advertisers from following dynamic bidding strategies that may reduce bid levels for the benefit of advertisers. In the next section, we study one such dynamic strategy that may lower the first position payment while increasing both advertisers’ payoffs.


Proposition 2 characterizes the symmetric equilibrium strategy profile that can generate BWCs.

advertisers outbid each other with certainty. This phase ends when an advertiser outbids the minimum relenting bid b and the war of attrition phase begins where advertisers outbid each other with positive probability. In this phase, each advertiser outbids the opponent hoping that the opponent will relent in the next period. The incentive to continue war of attrition arises, because an advertiser pays just l ? k for the first position in the next two periods after the opponent relents (see Fig. 4). However, the cost of continuing war of attrition increases as bids increase. If the first position bid becomes too high, i.e. as high as b þ k; the second position advertiser does not have any incentive to outbid. At the same time, the second position advertiser can not relent to l with probability one, otherwise the opponent may bid b þ k and always occupy the first position. Therefore, relenting always occurs with a probability less than one. Accordingly, at state b þ k) advertisers engage in an asymmetric war of attri(b; tion where X relents to l with probability 1-q and Y relents to l with probability 1-h.

where b* is defined in Proposition 2 If b þ k b b; Proposition 1, the following strategy profile {R, R} with R = RX(bX, bY) constitutes a Markov Perfect Equilibrium.

Example 1 Consider the following example. The parameters are given as follows: profit per click per period, s1 = s2 = $2.5; number of clicks per period, a1 = 20 and

4 Bidding war cycle equilibrium

8 > for bY b; bY þ k > > > > > bY þ k for b þ k bY \b and bX 6¼ bY k; > > > > > > bY þ k with probability lðbY Þ > > for b þ k bY b and bX ¼ bY k; > Y > > l with probability 1 l b ð Þ > > < b þ k for bY ¼ b and bX b 2k; X X Y R b ;b ¼ > > b þ k with probability h > > > for bY ¼ b and bX b þ k; > > l with probability 1 h > > > > > b with probability q > > > for bY b þ k and bX ¼ b; > > l with probability 1 q > > : b for bY b þ k and bX 6¼ b: l(b) for b þ k b b decreases linearly with b. Sketch of the proof The complete proof is included in Appendix A. We give a sketch of the proof as follows: We define the parameters b and b and show that the mixing To check for probabilities are well defined, if b þ k b b: whether (6) indeed forms an equilibrium, it is sufficient to show that there is no one-period profitable deviation [11]. There are two phases of the equilibrium path generated by the symmetric reaction function in Proposition 2 (see Fig. 4). A cycle begins when one of the advertisers relents to l. In the equilibrium, the opponent follows by bidding l ? k thereby starting the bidding war phase of the equilibrium in which

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a2 = 10; advertisers’ discount rate, d = 0.99; minimum bid, l = 10 cents; and grid size, k = 1 cent. Table 2 shows the symmetric BWC strategy of Proposition 2. Figure 4 shows simulated bids from the equilibrium reaction function in Table 2. At the end of the first cycle, X is successful in forcing Y to relent to 10 cents against X’s bid of 45 cents. X drops her bid to 11 cents as a response. In actual bidding situations this move is quite problematic; an advertiser in X’s role tends to delay dropping her bid, because she is only paying 11 cents as this is a second-price auction. If the advertiser’s delay is too long, other advertiser may react and continue outbidding. The equilibrium reaction function makes this deviation unprofitable for X.

Inf Technol Manag (2011) 12:1–16 Fig. 4 Simulated bids from the symmetric BWC equilibrium strategy of Table 2 in Example 1

7 $ 0.45 0.40 0.35

b 0.30 0.25 0.20 0.15 0.10

x x y y x x y y x x y y y y x x x x y y y y x x x x War of Attrition Phase y y y y x x x x y y y y x x x x y y y y x x x x y y y y x x x x y y y y x x x x y y y y x x x x y y y y Bidding War Phase x x x x y y y y x x x x y y y y x x x x y y y y x x x x y y y y x x x x Y relents X relents y y y y x x x x y y y y x x x x y y y x

0

10

20

30

40

50

60

t

x: X’s bids, y: Y’s bids Table 2 Symmetric BWC equilibrium strategy of Proposition 2 (in $) for Example 1 RX(bX, bY)

State (bX, bY)

bY ? 0.01

bY B 0.31

bY ? 0.01

0.32 B bY \ 2.56 and bX = bY-0.01

bY ? 0.01 with prob. l(bY) = 0.998–0.443(bY-0.32)

0.32 B bY B 2.56 and bX = bY-0.01

0.1

with prob. 1-l(bY)

2.57

with prob. h = 0.564

0.1

with prob. 1-h = 0.436

2.56

with prob. q = 0.00082

0.1

with prob. 1-q = 0.99918

2.56

bY = 2.56 and bX C 2.57 bY C 2.57 and bX = 2.56 bY C 2.57 and bX = 2.56

After X drops her bid, Y outbids X by only 1 cent. It is important to note that jump bidding is not advantageous to advertisers (shown in the proof of Proposition 2) because it escalates the bidding war phase. Advertisers do not relent early, either. It is optimal to outbid until bids reach 32 cents (also shown in the proof of Proposition 2). After 32 cents ðb þ 1 centÞ; advertisers use a mix strategy of outbidding and relenting to force the opponent to relent first. This is a war of attrition game in which staying in the game become more costly as time passes. Therefore, the probability of relenting increases. In Fig. 4, X relents first in the second cycle and a new cycle begins.

5 Search engine perspective A natural question for a search engine is how much value it extracts from advertisers under different bidding regimes. We have considered two such regimes: bidding b* which is

the weakly dominant bid in the one-shot simultaneousmove game in Proposition 1 and engaging in BWCs in which bids change in cycles between the minimum bid and a random relenting bid in Proposition 2. In our model, the payment at the second position is fixed at the minimum bid because the auction is a second-price auction. Therefore, the search engine is interested in the level of the average first position bid in order to compare the bidding regimes of Propositions 1 and 2. Figure 5 shows the simulated bids of Fig. 4 as well as b* and s1 (= s2) for Example 1. The most salient feature of Fig. 5 is that b* (= $1.3) and b ð¼ $0:31Þ are quite low relative to s1 (= $2.5). The average payment at the first position in BWCs bavg (= $0.301) is a little lower than b: We include the method of calculating this average bid for BWCs in Appendix B. If advertisers play their strategies in Proposition 1, the search engine captures only half of the value at the first position (b*/s1 = $1.3/$2.5 = 52%). This implies that this type of auction is not truthful—advertisers bid below their valuations in contrast with the traditional second-price auction for one item. If advertisers play a oneperiod game, they bid so that they are indifferent between the first position and the second position when their payment at the first position is equal to their bid. On the other hand, if advertisers engage in BWCs of Proposition 2, the search engine is only getting 12% of the value (bavg/s1 = $0.301/$2.5 = 12%), which is much lower than 52%. This example shows that BWCs may be a tacit collusion mechanism where advertisers keep bids quite low compared to their valuations and their weakly dominant bids. Even the maximum simulated bid in 1000 cycles ($0.85) does not reach the indifference bid b* (= $1.3). There are auction design mechanisms the search engine can employ to raise the bids. Fig. 6 shows that the first position bid in both the competitive and BWC equilibrium can be improved significantly by increasing the minimum

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8


Fig. 5 Simulated bids in Example 1 (in dark squares and circles) versus b* = $1.3, s1 = s2 = $2.5, and l = $0.10

Fig. 6 Simulated bids in Example 1 (in dark squares and circles) versus b* = $2.0 and s1 = $2.5 with l = $1.50

bid, l, from 10 cents to $1.50 in Example 1. b* increases to $2.00 (80% of s1). The average bids in BWCs increases to $1.626 (65% of s1), while b ¼ $1:63: The increase in the first and second position payments pushes the bids upwards. If we further increase l, the competitive and the BWC payments converge to the same value as l increases. A plausible way of achieving a higher l is to introduce a reserve price for the top two positions. Alternatively, the search engine can put an advertiser’s listing to a special position, if her bid exceeds a certain level. Corollary 1 provides a formal discussion of these results. Corollary 1 b*, defined in Proposition 1, is nondecreasing in the minimum bid, l. Let b ¼ l þ ð2t þ 1Þk: Then, within the range of parameter values that support the BWC equilibrium, if a1 [ a2, t is non-increasing in l. Proof

The proofs of Corollaries 1–4 are in Appendix A.

Corollary 1 indicates that an increase in the minimum bid may decrease the number of times advertisers outbid each other (2t ? 1). However, b is directly increasing with

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l since b ¼ l þ ð2t þ 1Þk: Therefore, the net impact of the increase in l and the decrease in t will determine the direction of change in b and in turn the average bid. In contrast, b* increases (more precisely non-decreasing) with l because it increases the second position payment and makes this position less desirable. Nevertheless, Fig. 6 shows that a poorly chosen l may lead to very low bids compared to advertiser valuations. Table 3 shows the effect of increasing the number of clicks per period at the second position, a2, from 10 to 19, which is only 1 click below a1 = 20. Both of the first position payments b* and bavg decrease drastically. The former drops to 9% and the latter drops to 6% of the value. In other words, improving the advertiser payoffs at the second position to almost match the first position payoff eliminates advertisers’ incentives to obtain the first position by raising their bids. Put differently, if the second position is more attractive to an advertiser, then she is willing to pay less for the first position. For example, to improve the incentives to bid for the first position, Google has a policy


9

Table 3 Summary of changes in bid levels in Example 1 in $ (as a percentage of s1 = s2) Quantity

Example 1*

a2 = 19

l = $1.50

d = 0.50

k = $0.10

b*

1.30 (52%)

2.00 (80%)

0.22 (9%)

1.30 (52%)

b

0.31 (12%)

1.63 (65%)

0.17 (7%)

1.01 (40%)

0.8 (32%)

0.301 (12%)

1.626 (65%)

0.158 (6%)

0.611 (24%)

0.678 (27%)

bavg

1.30 (52%)

*: s1 = s2 = $2.5; a1 ¼ 20; a2 ¼ 10; d = 0.99; l = $0.10; and k = $0.01

of promoting an advertiser’s first position listing from the right-hand side of the results page to a very visible position on the results page (below the search textbox).8 Corollary 2 provides a formal proof of these results. Corollary 2 b*, defined in Proposition 1, and within the range of parameter values that support the BWC equilibrium, b are non-increasing in s2 and a2 and non-decreasing in s1 and a1. Table 3 also shows that decreasing the discount factor, d, from 0.99 to 0.50 doubles the average first position payment in Example 1. The interpretation of this finding is as follows. Maskin and Tirole [11] show that the twoperiod commitment model is equivalent to a model in which advertisers wait for a random amount of time before reacting to the opponent’s bid. A higher discount factor implies that an advertiser waits longer on average because advertisers put more weight to the future earnings or they have a stronger belief that the BWCs will not break down in the future. A lower discount factor implies that advertisers bid more aggressively. Noting that b* does not change with d, Corollary 3 presents a formal statement of this result with respect to b: Corollary 3 Within the range of parameter values that support the BWC equilibrium, b is non-increasing in d. Finally, Table 3 shows that increasing k from 1 to 10 cents positively affects the average payment in BWCs in Example 1. The search engine may be better off with increasing the grid size, if they estimate that advertisers’ payoffs at the first and second positions are significantly different. If the first two positions are close, increasing k may in fact reduce average payments. The reason is that coarser grids fail to capture incremental value when valuations are close to each other. For example, an advertiser with s1 = $0.28 would bid only $0.20 for the first position with a grid size of $0.10 compared to a maximum bid of $0.28 with a grid size of $0.01. Corollary 4 provides a formal treatment. Corollary 4 b*, defined in Proposition 1, is non-increasing in the grid size k. Within the range of parameter values that support the BWC equilibrium, t is non-increasing in the grid size k. 8

This point was made by Hal Varian in his keynote speech at ACM Conference on Electronic Commerce 2006, Ann Arbor, MI.

On a coarser grid, b* may only decrease or stay the same but not increase. Grid size creates three effects on the BWC equilibrium. The first effect, which is stated in Corollary 4, is the increase in the cost of outbidding the rival, which may decrease the number of times advertisers outbid each other, 2t ? 1. The second effect is the direct positive effect on the value of b; because b ¼ l þ ð2t þ 1Þk: The third effect is that the bids traverse higher levels on a coarser bid grid. Accordingly, they may not capture incremental value. The net impact of these effects determines how average bids change in BWCs when k is increased.

6 Conclusion We provide a dynamic model of a second-price sponsored search auction introduced by Overture. We study two Markov Perfect Equilibria—one that produces stable bids and one that produces BWCs. We also study the sensitivity of the search engine’s payoff with respect to model parameters. Furthermore, we propose several ways of improving the search engine’s revenue. The search engine can make the first position more desirable by giving the first position better placement in the search results. Additionally, we show that a carefully chosen reserve price for top positions can significantly improve search engine’s revenues. We can also achieve this by restricting the number of slots. Finally, the choice of the grid size may improve search engine revenues.

Appendix A Proof of Proposition 2 Following Maskin and Tirole [12] and Eckert [4] the proof proceeds as follows. First, we define the parameters b and b and present the conditions that they must satisfy; then we show that the mixing probabilities l( ), h, q are well defined given b and b: And then, we show that no profitable one-period deviation exist for X. In other words, if X chooses a different bid than the one specified in the equilibrium strategy for one period and then follow the equilibrium strategy, this deviation must not be more profitable than its payoff when she follows the equilibrium strategy without any deviations. Since we consider stationary Markov strategies,

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10


Maskin and Tirole [11] Proposition 1 implies that this set of conditions is sufficient for proving the symmetric strategy profile in (6) forms an MPE.

continuation value from relenting in a cycle where the bidding war phase continues until the bid ðb þ 2kÞ þ k:

Expected continuation payoffs

Definition of b

We define three value functions to help with the exposition of the proof. Recall that there are two phases of the BWC equilibrium. In the bidding war phase, advertisers outbid each other to get the first position. In the ensuing war of attrition phase, each advertiser outbid the opponent with a probability less than one hoping the opponent would relent first. When the opponent relents, the advertiser enjoys the first position with the lowest payment (l ? k) for two periods. Let Vrelent(b) be X’s expected discounted total future payoff after X relents to l from state ð; bÞ;

b is the bid that marks the end of the bidding war phase. b is defined by the following conditions.10 1.

X weakly prefers {outbidding b k and then failing} to relenting. Formally, PX ðb; b kÞ þ dV fail ðbÞ V relent ðbÞ P1 ðbÞ þ dP2 þ d2 V relent ðbÞ V relent ðbÞ P1 ðbÞ þ dP2 V relent ðbÞ: 1 d2

V relent ðbÞ ¼ P2 þ dV X;Y ðl; bÞ and PX ðl; bÞ þ dPX ðl; l þ kÞ þ d2 PX ðl þ 2k; l þ kÞ þ þ d2tþ1 PX ðb k; bÞ ; 1 d2tþ2 P2 þ dP2 þ d2 P1 ðl þ 2kÞ þ þ d2tþ1 P2 : V relent ðbÞ ¼ 1 d2tþ2

V relent ðbÞ ¼

ðA1Þ

for b ¼ l þ ð2t þ 1Þk with t a positive integer.9 V relent ðbÞ represents X’s expected continuation payoff from relenting. Next, we define the continuation payoff when an advertiser fails at forcing the opponent to relent. That is, V fail ðbÞ represents X’s expected continuation payoff when the opponent Y outbids X’s bid bX, and then X relents. V fail ðbÞ ¼ PX bX ; bX þ k þ dV relent ðbÞ ðA2Þ V fail ðbÞ ¼ P2 þ dV relent ðbÞ

2.

Similarly, we now define the continuation payoff when an advertiser is successful at forcing the opponent to relent. Accordingly, V success ðbÞ is the X’s expected continuation payoff when Y relents against X’s bid bX. V success ðbÞ ¼ PX bX ; l þ dPX ðl þ k; lÞ þ d2 V X;Y ðl þ k; lÞ

P1 ðb þ kÞ þ dP2 P1 ðbÞ þ dP2 \V relent ðbÞ : 2 1d 1 d2

V success ðbÞ ¼ P1 ðl þ kÞ þ dP1 ðl þ kÞ þ d2 V X;Y ðl þ k; lÞ

b is the bid that marks the end of the war of attrition phase. b is defined by the following conditions.11

ðA3Þ for bX [ l. Note that V success ðbÞ [ V fail ðbÞ: The expected continuation payoff functions V relent ðbÞ; V fail ðbÞ; V success ðbÞ depend on b because a different parameter value represents a different BWC. For example, V relent ðb þ 2kÞ represents the 9

It can be shown that b ¼ l þ ð2t þ 1Þk is equivalent to choosing b ¼ l þ 2tk; therefore, we only consider the former case throughout the paper.

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X prefers relenting to {outbidding b and then failing}. Formally, V relent ðbÞ [ PX ðb þ k; bÞ þ dV fail ðbÞ V relent ðbÞ [ P1 ðb þ kÞ þ dP2 þ d2 V relent ðbÞ V relent ðbÞ [

P1 ðb þ kÞ þ dP2 : 1 d2

Putting both of the conditions together produces the equilibrium condition (A4). ðA4Þ

Definition of b

10

It is important to understand the distinction between this definition and the equilibrium strategy. Note that in equilibrium, X outbids b with probability one. This does not violate the equilibrium conditions, because X is indifferent between outbidding b and relenting since X does not always fail after outbidding b; that is, Y relents with a positive probability against b þ k: See Equation (A6). 11 It is also important to understand the distinction between this definition and the equilibrium strategy. In equilibrium X is indifferent between


1.

2.

11

X weakly prefers {outbidding b and then succeeding} to relenting. Formally, PX ðb þ k; bÞ þ dV success ðbÞ V relent ðbÞ: X prefers relenting to {outbidding b þ k and then succeeding}. Formally, V relent ðbÞ [ PX ðb þ 2k; b þ kÞþ dV success ðbÞ:

Putting these conditions together produces the following condition that defines b PX ðb þ k; bÞ þ dV success ðbÞ V relent ðbÞ [ PX ðb þ 2k; b þ kÞ þ dV success ðbÞ;

In order to show that l(b) decreases linearly with b, consider l(b ? k), from (A6) it satisfies V relent ðbÞ ¼ P1 ðb þ kÞ þ lðb þ kÞdV fail ðbÞ þ ð1 lðb þ kÞÞdV success ðbÞ Subtracting this from (A6), we obtain P1 ðbÞ P1 ðb þ kÞ þ flðbÞ lðb þ kÞg d V fail ðbÞ V success ðbÞ ¼ 0; P1 ðb þ kÞ P1 ðbÞ dfV success ðbÞ V fail ðbÞg a1 k ¼ : dfV success ðbÞ V fail ðbÞg

lðb þ kÞ lðbÞ ¼

P1 ðb þ kÞ þ dV success ðbÞ V relent ðbÞ [ P1 ðb þ 2kÞ þ dV success ðbÞ:

ðA5Þ

ðA7Þ

Because V success ðbÞ [ V fail ðbÞ and V success ðbÞ and V fail ðbÞ are independent of b, l(b) decreases linearly with b.

War of attrition Definition of l(b)

Definition of q

X In the war of attrition phase, in which b þ k b b; and Y outbid each other with some positive probability hoping the opponent will relent. For X to use a mixed strategy, X and Y must be indifferent between continuing the bidding war and relenting throughout the war of attri l(b) satisfies: tion phase. Thus, for b þ k b b; Y

V relent ðbÞ ¼ PY ðb k; bÞ þ lðbÞdV fail ðbÞ þ ð1 lðbÞÞdV success ðbÞ; V relent ðbÞ ¼ P1 ðbÞ þ lðbÞdV fail ðbÞ þ ð1 lðbÞÞdV success ðbÞ:

ðA6Þ

(A6) indicates that X bids b ? k at state (b-k, b) with probability l(b) to make Y indifferent between continuing the war of attrition and relenting at state (b-k, b-2 k). Next, we show that the probability of continuing the war of is well-defined, i.e. attrition, l(b) for b þ k b b; 0 B l(b) B 1, due to the conditions (A4) and (A5). we must have When lðbÞ ¼ 1 for b þ k b b; relent fail ðbÞ P1 ðbÞ þ dV ðbÞ: This holds because in V deriving (A4), we use the condition V relent ðbÞ [ P1 ^ for ðb þ kÞ þ dV fail ðbÞ: This condition and P1 ðbÞ\P1 ðbÞ b [ b^ imply that V relent ðbÞ [ P1 ðbÞ þ dV fail ðbÞ: we must have V relent When l(b) = 0 for b þ k b b; success ðbÞ P1 ðbÞ þ dV ðbÞ: This follows from the left-hand side of (A5), P1 ðb þ kÞ þ dV success ðbÞ V relent ðbÞ, and ^ for b [ b: ^ P1 ðbÞ\P1 ðbÞ

In the event that Y’s bid reaches b þ k, X has no incentive to outbid Y. However, X cannot relent with probability one, since Y would always bid b þ k and always occupy the first position. In fact, relenting with probability one can never be part of a bidding war cycle equilibrium. Therefore, X uses a mixed strategy between bidding b and b þ kÞwith probability q relenting to l; X bids b at state ðb; and l with probability 1 - q to make Y indifferent between b kÞ and choosing b þ k and l (relenting) at states ðb; Y Y ðb; b Þ for b b þ k: Hence, q satisfies: b þ kÞ þ qd PY ðb; b þ kÞ þ dV relent ðbÞ V relent ðbÞ ¼ PY ðb; þ ð1 qÞdV success ðbÞ; V relent ðbÞ ¼ P1 ðb þ kÞ þ qd P1 ðb þ kÞ þ dV relent ðbÞ þ ð1 qÞdV success ðbÞ:

ðA8Þ

For q to be well-defined, we must have V relent ðbÞ P1 ðb þ kÞ þ dV success ðbÞ; when q = 0. This directly follows from the left-hand side of (A5). Likewise, when q = 1, we must have V relent ðbÞ P1 ðb þ kÞ þ dP1 ðb þ kÞ þ d2 V relent ðbÞ; V relent ðbÞ

P1 ðb þ kÞ þ dP1 ðb þ kÞ: 1 d2

Since b þ k b þ k and from the left-hand side of (A4), this condition is guaranteed to hold, if P1 ðb þ kÞ P2

Footnote 11 continued outbidding b and relenting as X does not always succeed: Y may stay See Equation (A8). at b with probability q [ 0 at state ðb þ k; bÞ:

Because P1 ðb þ kÞ\P2 ; a sufficient condition for this inequality is b þ k b þ k b b ðA9Þ

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12


Definition of h For an advertiser to mix between b and l, she must be indifferent between these two bids. As a result, we need another mixed strategy for establishing this indifference. X bids b þ k at states ðbX ; bÞ for bX b þ k with probability h and l with probability 1-h to make Y indifferent between choosing b and l at state ðb þ k; bÞ: Hence, h satisfies: V relent ðbÞ ¼ PY ðb þ k; bÞ þ hd PY ðb þ k; bÞ þ dV relent ðbÞ þ ð1 hÞdV success ðbÞ;

8 Y > >b þ k < bY þ k X X Y R b ;b ¼ >b þ k > : b

for bY ¼ b; for b þ k bY \b and bX 6¼ bY k; for bY ¼ b and bX b 2k; for bY b þ k and bX 6¼ b:

Proof of Lemma A1 We prove the claim for each case one by one. The expected continuation payoff from RX ðbX ; bY Þ ¼ bY þ k for bY ¼ b is PX ðb þ k; bÞ þ lðb þ kÞdV fail ðbÞ þ ð1 lðb þ kÞÞ

V relent ðbÞ ¼ P2 þ hdV fail ðbÞ þ ð1 hÞdV success ðbÞ: ðA10Þ

dV success ðbÞ ¼ P1 ðb þ kÞ þ lðb þ kÞdV fail ðbÞ

For h to be well-defined, we must have V relent ðbÞ P2 þ dV success ðbÞ; when h = 0. This is guaranteed to hold, if (A9) holds, due to the left-hand side of (A5). Likewise, when h = 1, we must have

þ ð1 lðb þ kÞÞdV success ðbÞ ¼ V relent ðbÞ:

V relent ðbÞ P2 þ dV fail ðbÞ; P2 þ dP2 : V relent ðbÞ 1 d2 From the left-hand side of (A4), this is guaranteed to hold, if P1 ðb þ kÞ P2 Because P1 ðb Þ P2 ; a sufficient condition for this inequality is b þ k b :

ðA11Þ

Putting (A9) and (A11) together produces the sufficient condition in Proposition 2 b þ k b b:

from (A6). The expected continuation payoff from RX ðbX ; bY Þ ¼ Y b þ k for b þ k bY \b and bX 6¼ bY k is PX bY þ k; bY þ l bY þ k dV fail ðbÞ þ 1 l bY þ k dV success ðbÞ ¼ P1 bY þ k þ l bY þ k dV fail ðbÞ þ 1 l bY þ k dV success ðbÞ ¼ V relent ðbÞ: from (A6). The expected continuation payoff from RX ðbX ; bY Þ ¼ b þ k for bY ¼ b and bX b 2k is PX ðbþ k; bÞ þ qd PX ðbþ k; bÞ þ dV relent ðbÞ þ ð1 qÞ dV success ðbÞ ¼ P1 ðbþ kÞ þ qd P1 ðbþ kÞ þ dV relent ðbÞ þ ð1 qÞdV success ðbÞ ¼ V relent ðbÞ:

all the We have shown that given b þ k b b; parameters in Proposition 2 are well defined.

from (A8). The expected continuation payoff from RX ðbX ; bY Þ ¼ b for bY b þ k and bX 6¼ b is

One-period deviation conditions

b þ kÞ þ hd PX ðb; b þ kÞ þ dV relent ðbÞ þ ð1 hÞ PX ðb;

In this section, we prove that there is no one-period profitable deviation and thus the symmetric strategy profile of Proposition 2 is indeed a Markov perfect equilibrium (Maskin and Tirole [11]). We consider each possible deviation and show that this deviation is not profitable as a response to any bY . The static symmetric reaction function (6) consists of two types of bids: the bids on the equilibrium path and the bids that return the system to the equilibrium path from offequilibrium states. In Lemma A1, we show that all the bids that return the system to the equilibrium path from offequilibrium states are equivalent to relenting. Lemma A1 The expected continuation payoffs from the following bids at the corresponding states are equal to the expected continuation payoff from relenting to l, V relent ðbÞ:

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dV success ðbÞ ¼ P2 þ hdV fail ðbÞ þ ð1 hÞdV success ðbÞ ¼ V relent ðbÞ: from (A10).

Q.E.D.

We proceed by showing that when an advertiser relents, she relents to l not to a higher bid. We use R~X to represent deviation bids. Lemma A2

RX ðbX ; bY Þ ¼ l for bY b þ k.

Proof of Lemma A2 Deviation A2.1 first, show that deviation. Then,

There are four possible deviations.

R~X ðbX ; bY Þ ¼ b where l\b b 2k:We, R~X ðbX ; bY Þ ¼ l þ k is not a profitable we show that R~X ðbX ; bY Þ ¼ l þ k is a


13

better response than R~X ðbX ; bY Þ ¼ l þ 2k: Hence, the result follows by induction on k. The difference between the expected continuation payoff from relenting, RX ðbX ; bY Þ ¼ l; and R~X ðbX ; bY Þ ¼ l þ k is X

X

2tþ1

We, now, show that there is no profitable one-period deviation from the bidding war phase of the equilibrium in Lemma A3. Lemma A3

X

P ðl; bÞ þ dP ðl; l þ kÞ þ þ d P ðb k; bÞ X 2tþ2 relent Y V ðbÞ P l þ k; b þ dPX ðl þ k; l þ 2kÞ þd þ þ d2t PX ðb; b kÞ þ d2tþ1 PX ðb; b þ kÞ þd2tþ2 V relent ðbÞ ¼ d2 ðP1 ðl þ 2kÞ P1 ðl þ 3kÞÞ þ : þd2t ðP1 ðb kÞ P1 ðbÞÞ [ 0: The difference between the expected continuation payoff from R~X ðbX ; bY Þ ¼ l þ k and R~X ðbX ; bY Þ ¼ l þ 2k is PX ðl þ k; bÞ þ dPX ðl þ k; l þ 2kÞ þ þ d2t PX ðb; b kÞ þ d2tþ1 PX ðb; b þ kÞ þ d2tþ2 V relent ðbÞ PX l þ 2k; bY þ dPX ðl þ 2k; l þ 3kÞ þ þ d2t1 PX ðb k; bÞ þd2t V relent ðbÞ ¼ d2 ðP1 ðl þ 3kÞ P1 ðl þ 4kÞÞ þ þ d2t2 ðP1 ðb 2kÞ P1 ðb kÞÞ þ d2t ðP1 ðbÞ þ dP2 1 d2 V relent ðbÞ [ 0: The inequality follows from the right-hand side of (A4).

RX ðbX ; bY Þ ¼ bY þ k for bY b:

Proof of Lemma A3 There are four possible deviations. We need to consider two cases when bY l is an odd ðbY ¼ l þ ð2r þ 1ÞkÞ and even ðbY ¼ l þ 2rkÞ multiple of k. r is a positive integer in the rest of the proof for Deviations A3.1 and A3.2. Deviation A3.1 R~X ðbX ; bY Þ ¼ l: In Deviation A2.1 we show that if an advertiser drops her bid below the opponent’s bid, the best bid is l. Therefore, we only consider relenting as a deviation by bidding below Y’s bid. Case A3.1.1 bY ¼ l þ ð2r þ 1Þk For RX ðbX ; bY Þ ¼ bY þ k (optimal reaction function), X’s expected continuation payoff is PX bY þ k; bY þ d PX bY þ k; bY þ 2k þ þ d2t2r2 PX ðb k; b 2kÞ þ d2t2r1 PX ðb k; bÞ þ d2t2r V relent ðbÞ ¼ P1 bY þ k þ dP2 þ

Deviation A2.2 R~X ðbX ; bY Þ ¼ b where b k b bY k: The difference between the expected continuation payoff from relenting and this deviation is V relent ðbÞ PX b; bY þ dPX ðb; b þ kÞ þ d2 V relent ðbÞ ¼ 1 d2 V relent ðbÞ ðP2 þ dP2 Þ 0:

The continuation payoff of R~X ðbX ; bY Þ ¼ l is V relent ðbÞ: Thus, the difference is

The inequality follows from (A11).

P1 bY þ k þ dP2 þ þ d2t2r2 P1 ðb kÞ

Deviation A2.3 R~X ðbX ; bY Þ ¼ b; where bY þ k\b b: The difference between the expected continuation payoff from relenting and this deviation is V relent ðbÞ PX b; bY þ dPX ðb; b þ kÞ þ d2 V relent ðbÞ ¼ 1 d2 V relent ðbÞ P1 bY þ k þ dP2 [ 0: This inequality follows from the left-hand side of (A4). Deviation A2.4 R~X ðbX ; bY Þ ¼ b where bY þ k\b and b b þ k:The difference between the expected continuation payoff from relenting and this deviation is V relent ðbÞ PX b; bY þ dPX ðb; bÞ þ d2 V relent ðbÞ ¼ 1 d2 V relent ðbÞ P1 bY þ k þ dP1 ðb þ kÞ 0: This inequality follows from the left-hand side of (A4) and (A9). The result follows from Lemma A1, (A6), (A8), and (A10). Q.E.D.

þ d2t2r2 P1 ðb kÞ þ d2t2r1 P2 þ d2t2r V relent ðbÞ:

þ d2t2r1 P2 þ d2t2r V relent ðbÞ V relent ðbÞ 0 P1 ðbY þ kÞ þ dP2 þ þ d2t2r2 P1 ðb kÞ þ d2t2r1 P2 1 d2t2r V relent ðbÞ: This holds because P1 ðbY þ kÞ þ dP2 þ þ d2t2r2 P1 ðb kÞ þ d2t2r1 P2 1 d2t2r P1 ðbÞ þ dP2 þ þ d2t2r2 P1 ðbÞ þ d2t2r1 P2 [ 1 d2t2r P1 ðbÞ þ dP2 ¼ V relent ðbÞ: 1 d2 The last inequality follows from (A4). Case A3.1.2 bY ¼ l þ 2rk

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14


For RX ðbX ; bY Þ ¼ bY þ k; X’s expected continuation payoff is PX bY þ k; bY þ d PX bY þ k; bY þ 2k þ þd

2t2r

X

P ðb; b kÞ þ d

2t2rþ1

X

P ðb; b þ kÞ þ d2t2rþ2 V relent ðbÞ ¼ P1 bY þ k þ dP2 þ þ d2t2r P1 ðb kÞ þ d2t2rþ1 P2 þ d2t2rþ2 V relent ðbÞ: The difference between this quantity and V relent ðbÞ is P1 bY þ k þ dP2 þ þ d2t2r P1 ðb kÞ þ d2t2rþ1 P2 þ d2t2rþ2 V relent ðbÞ V relent ðbÞ 0 P1 ðbY þ kÞ þ dP2 þ þ d2t2r P1 ðb kÞ þ d2t2rþ1 P2 d2t2rþ2 relent V ðbÞ: This holds, similarly from (A4) P1 ðbY þ kÞ þ dP2 þ þ d2t2r P1 ðb kÞ þ d2t2rþ1 P2 d2t2rþ2 P1 ðbÞ þ dP2 þ þ d2t2r P1 ðbÞ þ d2t2rþ1 P2 [ d2t2rþ2 P1 ðbÞ þ dP2 ¼ V relent ðbÞ: 1 d2 Deviation A3.2 R~X ðbX ; bY Þ ¼ b where bY þ k\b\b k: This deviation stands for jump bidding – i.e. outbidding the rival more than k in the bidding war.We show that X chooses RX ðbX ; bY Þ ¼ bY þ k rather than RX ðbX ; bY Þ ¼ bY þ 2k; then the result follows by induction. Case A3.2.1 bY ¼ l þ ð2r þ 1Þk For RX ðbX ; bY Þ ¼ bY þ k; X’s expected continuation payoff is PX bY þ k; bY þ d PX bY þ k; bY þ 2k þ þ d2t2r2 PX ðb k; b 2kÞ þ d2t2r1 PX ðb k; bÞ þ d2t2r V relent ðbÞ: For RX ðbX ; bY Þ ¼ bY þ 2k; X’s expected continuation payoff is PX bY þ 2k; bY þ d PX bY þ 2k; bY þ 3k þ

This inequality clearly holds. Case A3.2.2 bY ¼ l þ 2rk For RX ðbX ; bY Þ ¼ bY þ k; X’s expected continuation payoff is PX bY þ k; bY þ dPX bY þ k; bY þ 2k þ þ d2t2r PX ðb; b kÞ þ d2t2rþ1 PX ðb; b þ kÞ þ d2t2rþ2 V relent ðbÞ: For RX ðbX ; bY Þ ¼ bY þ 2k, X’s expected continuation payoff is PX bY þ 2k; bY þ dPX bY þ 2k; bY þ 3k þ þ d2t2r2 PX ðb k; b 2kÞ þ d2t2r1 PX ðb k; bÞ þ d2t2r V relent ðbÞ: The difference is d2 P1 bY þ 3k P1 bY þ 4k þ d4 P1 bY þ 5k P1 bY þ 6k þ þ d2t2r P1 ðbÞ þ dP2 1 d2 V relent ðbÞ [ 0: This inequality follows from the right-hand side of (A4).By induction, X does not choose b ¼ bY þ jk\b k; j [ 1 since P1 is a decreasing function. Deviation A3.3 R~X ðbX ; bY Þ ¼ b where bY þ k\b and This deviation is equivalent to Deviation b k b b: A2.3. Deviation A3.4 R~X ðbX ; bY Þ ¼ b where bY þ k\b and b b þ k: This deviation is equivalent to Deviation A2.4. Q.E.D. We have shown that there is no profitable one-period deviation from the equilibrium reaction function. This completes the proof of Proposition 2. Q.E.D. Lemma A4 Let b ¼ l þ ð2t þ 1Þk; then V relent ðbÞ V relent ðb 2kÞ 0 implies that a1 k 2t þ d4 ð2t 4Þ þ d6 ð2t 6Þ þ þ d2t2 2 a1 ðs1 lÞ a2 ðs2 lÞ

þ d2t2r2 PX ðb; b kÞ þ d2t2r1 PX ðb; b þ kÞ

Proof of Lemma A4 can write:

þ d2t2r V relent ðbÞ

V relent ðbÞ

The difference is d2 P1 bY þ 3k P1 bY þ 4k þ d4 P1 bY þ 5k P1 bY þ 6k þ þ d2t2r2 ðP1 ðb kÞ P1 ðbÞÞ [0

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Using (A1), for b ¼ l þ ð2t þ 1Þk we

P2 þdP2 þd2 P1 ðlþ2kÞþþd2t P1 ðbkÞþd2tþ1 P2 1d2tþ2 relent V ðb2kÞ ¼

¼

P2 þdP2 þd2 P1 ðlþ2kÞþþd2t2 PX ðb3kÞþd2t1 P2 1d2t


15

This contradicts with f ðtL þ 1Þ [ gðlL Þ: This implies that tH must be at most as large as tL, tH B tL. Q.E.D.

1 Let D¼1d12tþ2 and E¼ 1d 2t [D; then we have

V relent ðbÞ V relent ðb 2kÞ ¼ ðD EÞðP2 þ dP2 þd2 P1 ðl þ 2kÞ þ þ d2t2 P1 ðb 3kÞ þ d2t1 P2

þ Dd2t ðP1 ðb kÞ þ dP2 Þ V relent ðbÞ V relent ðb 2kÞ 0 implies

E 1 P2 þ dP2 þ d2 P1 ðl þ 2kÞ þ D þd2t2 P1 ðb 3kÞ þ d2t1 P2 d2t ðP1 ðb kÞ þ dP2 Þ It is easy to show that E d2t ð1 d2 Þ d2t 1¼ ¼ 2t 2 4 D 1 þ d þ d þ þ d2t2 1d From the last inequality and the last equation, we get d2 ðP1 ðl þ 2kÞ P1 ðb kÞÞ þ d4 ðP1 ðl þ 4kÞ P1 ðb kÞÞ þ þ d2t2 ðP1 ðb 3kÞ P1 ðb kÞÞ P1 ðb kÞ P2 :

Proof of Corollaries 2, 3 and 4 The same proof method used in Corollary 1’s proof easily applies to Corollaries 2, 3 and 4. Appendix B Calculation of the average first position bid on the BWC Equilibrium path We define the following variables to help the exposition: any bid b ¼ l þ nk and the equilibrium parameters b ¼ l þ nk and b ¼ l þ nk: Moreover, we define the relenting bid, br ¼ l þ nr k; as the maximum value of bids that can be reached in a BWC. Note that this is a random variable. We first characterize the distribution of the relenting bid and then calculate the payment per period for each realization of the relenting bid. 1.

The discrete distribution of the relenting bid, br ; is: Prðbr ¼ bÞ 8 0 for b\b þ k or b [ b þ k; > > > > > > nn1 > Y < ð1 lðbÞÞ lðb þ ikÞ for b þ k b b; ¼ i¼1 > > > > nQ n > > > lðb þ ikÞ for b ¼ b þ k :

2.

We define the function Sðbr Þ as the first position bid function for each realization of br : then the average first position bid is If b þ k br b;

n P 1 SðbÞ ¼ nr þ1 l þ k þ l þ ik for b þ k b b

Substituting in the one-period payoff function (2), a1 k d4 ð2t 4Þ þ d6 ð2t 6Þ þ þ d2t2 2 a1 ðs1 ðl þ 2tkÞÞ a2 ðs2 lÞa1 k 2t þ d4 ð2t 4Þ þd6 ð2t 6Þ þ þ d2t2 2 a1 ðs1 lÞ a2 ðs2 lÞ: ðA12Þ Q.E.D. Proof of Corollary 1 First, we prove the result on b: Note that the minimum relenting bid b can also be characterized by the following inequalities: V relent ðbÞ V relent ðb 2kÞ 0 [ V relent ðb þ 2kÞ ðA13Þ V relent ðbÞ: Let f (t) be the left-hand side of (A12) as a function of t and g (l) be the right-hand side of (A12) as a function of l. Note that f (t) is increasing in t and is g (l) decreasing in l, because a1 [ a2.Let b ¼ lL þ ð2tL þ 1Þk be the minimum relenting bid for lL, and b0 ¼ lH þ ð2tH þ 1Þk be the minimum relenting bid for lH (lL \ lH). Thus, we know that the following inequalities hold. f ðtL Þ gðlL Þ; f ðtL þ 1Þ [ gðlL Þ and ðtH Þ gðlH Þ; f ðtH þ 1Þ [ gðlH Þ from (A13). Because g(l) decreasing in l, gðlH Þ\gðlL Þ: Suppose tH C tL ?1, then it follows that f ðtL þ 1Þ gðlH Þ\gðlL Þ ) f ðtL þ 1Þ\gðlL Þ

i¼1

a.

i¼1

If br ¼ b þ k; then the average first position bid is: ! nþ1 X 1 lþkþ ðl þ ikÞ ð1 qÞ n þ 2 i¼1 ! nþ1 X 1 lþkþ þ qð1 hÞ ðl þ ikÞ þ ðb þ kÞ n þ 3 i¼1 ! nþ1 X 1 lþkþ þ qhð1 qÞ ðl þ ikÞ þ 2ðbþ kÞ n þ 4 i¼1 ! nþ1 X 1 2 lþkþ þ q hð1 hÞ ðl þ ikÞ þ 3ðb þ kÞ þ n þ 5 i¼1

b.

Hence, the average first position bid can be written as:

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16


! nþ1 X 1 lþkþ Sðb þ kÞ ¼ ð1 qÞ ðl þ ikÞ þ n þ 2 i¼1 1 X

j

qb 2 c hb2c ð1 qÞI2 ðjÞ ð1 hÞ1I2 ðjÞ jþ1

j¼1 nþ1 X

1 n þ 2 þ j

!

lþkþ ðl þ ikÞ þ jðb þ kÞ i¼1 1; j is an even number, where I2 ðjÞ ¼ 0; j is an odd number:

Since each individual term in the last summation expression tends to zero as j increase, we only calculate the sum of the first 10 terms in this expression. Finally, we can write the overall average first position bid as: S¼

nnþ1 X

Sðb þ ikÞ Prðbr ¼ b þ ikÞ

i¼1

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